A transformation semigroup (ts) X = (Q,, S,) in this paper consists of a finite set of states Qx and a subsemigroup of transformations S, of PF(Q,), the monoid of all partial functions on Q, with composition as multiplication. For n > 1, ii’ denotes the transformation monoid (tm) with n states and with the n constant maps on those n states (along with the identity function) as transformations. ...

We investigate the length `(n, k) of a shortest preset distinguishing sequence (PDS) in the worst case for a k-element subset of an n-state Mealy automaton. It was mentioned by Sokolovskii [18] that this problem is closely related to the problem of finding the maximal subsemigroup diameter `(Tn) for the full transformation semigroup Tn of an n-element set. We prove that `(Tn) = 2 n exp{ √ n 2 l...

Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U ≈ V if there exists a finite subset F of ΩΩ such that the subgroup generated by U and F equals that generated by V and F . The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. We study the notions of relative...

We give, in terms of (generalized) cyclic types, a criterion for two elements of Tn, PT n, IS(N), Bn, PBn and Cn to be conjugate with respect to the conjugation defined as the transitive closure of the xy ∼ yx relation.

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective cor...

In this paper, we introduce generalized adjoint semigroups (GA-semigroups) of a ring R. We construct generalized adjoint semigroups on a ring R by means of bitranslations of R. It is shown that GA-semigroups of a π-regular ring are π-regular. As an application we deduce that in any ring, idempotents can be lifted modulo π-regular ideals. GA-semigroups containing idempotents are described in ter...

We give an upper bound of n((n−1)!−(n−3)!) for the possi5 ble largest size of a subsemigroup of the full transformational semigroup 6 over n elements consisting only of nonpermutational transformations. As 7 an application we gain the same upper bound for the syntactic complex8 ity of (generalized) definite languages as well. 9

In this paper a certain class of polyhedral semigroups which has a presentation $$ is examined‎. ‎The completeness of the presentation and solvability of word problem of this class of semigroups is determined‎. ‎Moreover the combinatorial distance between two words is determined‎.